A fundamental characteristic of mankind""s use of the low-Earth-orbit (LEO) environment is that devices placed there usually result in the generation of orbital debris as a by-product. When payloads are launched, operational debris pieces and rocket bodies are also often placed in the environment. In some cases, these objects have not remained on orbit as inert hulks; spontaneous disintegrations have often replaced a single large piece of debris piece with up to hundreds of smaller pieces. In fact, approximately 50% of all objects currently tracked were generated by fragmentations of one type or another.
Even payloads themselves tend first to become derelicts before they decay from the environment; presently approximately four out of every five such objects are useless hazards to navigation. From this it may be concluded that the average orbital life of the typical payload is, at least, several times greater than its functional life. Finally, although not yet a significant contributor to the buildup of debris in the LEO environment, collisions may become more frequent as the environment becomes increasingly crowded.
Taken together, approximately 95% of all mass in LEO is trash, and a host of smaller, yet dangerous, objects are suspected to be present. With the exception of very few cases of retrieval (e.g., Long Duration Exposure Facility, or LDEF), the only debris removal mechanism operating in the environment is drag due to the residual atmosphere at LEO altitudes. Even this mechanism was shown to be ineffective above an altitude of 750 km.
Since the continued use of the LEO environment is assured during the upcoming era of International Space Station (ISS) and the anticipated proliferation of LEO constellations of communication satellites (comsat), navigation satellites and other high-value systems, a present desideratum would be the development of methods to assess the impact of mankind""s activities on the environment and, in turn, the impact of the evolving environment on mankind""s further use. Presumably, if successful in this pursuit, the user community will be able to determine, with sufficient lead-time, what activities and policies are most likely to lead to a stable and desirable environment in the long term. In addition to large-scale LEO debris environment issues, predictive risk assessment models are urgently needed relative to high-value assets, whether individual in nature (e.g., ISS) or operational collectives (e.g., comsat constellations).
The utilization of space is increasing as commercial, military, government, research, and academic agencies discover new ways to exploit the use of this environment. With the increase in the numbers of satellites and debris orbiting Earth comes the increase in importance of protecting the safety of manned and unmanned space-based assets. This explosive growth rate is expected to increase with the deployment of large satellite constellations, both military and commercial. The advent of large low-Earth-orbit (LEO) satellite constellations presents a significant new issue for the orbital-debris environment; this presence of large numbers of commercial satellites is a new phenomenon for modelers of space debris.
Policies, specifically NASA Management Instruction 1700.8 and the Department of Defense Space Policy, dictate NASA and Department of Defense space-faring agencies to strive to minimize or reduce accumulation of space debris consistent with mission requirements and costs. All commercial activities are subject to the Department of Transportation (DOT) Office of Commercial Space Transportation""s regulations requiring them to address safety issues with respect to launches, including the risks associated with orbital debris and on-orbit proliferation.
Results of the National Science and Technology Council Committee on Transportation Research and Development Interagency Report on Orbital Debris (1995) concluded:
xe2x80x9cThere is a need to characterize the orbital debris environment, even when observations are not practical, such as when the size or altitude of objects makes measurements difficult. Modeling, then, is required to combine existing measurements and theory in such a way that predictions can be made. Several types of models are required to make these predictions:
(1) A model to describe future launches, the amount of debris resulting from these launches, and the frequency of accidental or intentional explosions in orbit (traffic model).
(2) A model to describe the number of fragments, fragment size, and velocity distribution of ejected fragments resulting from a satellite explosion or collision (breakup models).
(3) A model which will make long-term predictions of how debris orbits will change with time (propagation model).
(4) A model which predicts collision probabilities for spacecraft (flux or risk model).
(5) A model which predicts hazards in the near term from a breakup event.
The Particle-in-a-Box (PIB) model of the present invention was developed specifically to address concerns (1) through (5) stated above. Results show the model provides an effective predictive tool to address the above concerns. A concluding recommendation of the interagency report stated that NASA and Department of Defense should continue current activities in orbital-debris research with particular attention to those orbits where critical national security payloads may be located, including International Space (ISS) and telecommunication constellations. This PIB model accomplishes this task with a limited object-size resolution. Recommendations also stressed the importance of focused studies on debris and emerging LEO systems. This invention provides a commercially available user-friendly space environment modeling package much needed by government agencies and commercial entities to address these policies and requirements.
The original PIB model was a single-particle, single-stratum averaged treatment of LEO, capable of global evolutionary and stability analysis. Without sacrificing its capabilities, the model resolution is increased by more than an order of magnitude from the original PIB model, resulting in substantial pay-off. The increased ability of the model to accept detailed phenomenological data represents a quantum leap in modeling applicability, as was shown by analysis of impact-risk-assessment studies of high-value assets such as the International Space Station and constellation, with cataloged objects (40 centimeter objects and larger).
An exemplary embodiment addresses impact risks significant for all space-based assets, including astronauts on extra-vehicular activity (EVA). This increase in fidelity improves the global (evolutionary and stability) analysis, which the PIB was originally developed to perform.
In developing a mathematical model of any evolving system, one must first choose a relevant parameter as the xe2x80x9cstatexe2x80x9d quantity. In developing the current model, the number of objects resident in the LEO environment at any given time was selected. The primary reason for this choice is that if an object can be seen, it can be countedxe2x80x94the number of objects on orbit is a direct observable subject, of course, to an appreciation of possible incompleteness, especially at higher altitudes and smaller sizes. The basic equation of the model is presented as follows:
{dot over (N)}=A+BN+CN2xe2x80x83xe2x80x83(1)
Where: N=number of objects on orbit
{dot over (N)}=time rate of change of the number of objects
A=deposition coefficient
B=removal coefficient
C=collision coefficient
The form of the equation follows from the assumptions that: (1) deposition reflects the rate at which users of the LEO environment choose to populate it with new objects; (2) decay due to atmospheric drag and/or random (debris sweeper) removal may be represented as a finite probability per unit time of the decay of any given LEO object; and (3) the theory for collisions between members of the population may be developed along a line of reasoning similar to that for collisions between particles in a gas whose mean free paths between collisions may be calculated. Each of these coefficients will be described in turn with clear illumination of their phenomenological character.
It is a historical fact that objects are launched into the LEO environment and that examination of the available data (e.g., NASA Satellite Situation Reports) will reveal that it is not unusual, on the average, for more than one object to be placed in LEO per launch. This activity deposits objects, mass and collision xe2x80x9ctargetxe2x80x9d area on orbit. Launch activity is a planned, intelligent activity and the typical number of objects deployed per launch is a reflection of policies, procedures and mission requirements.
Furthermore, it has been observed that some objects, initially intact, later fragment on orbit. As a result of such accidents, no additional mass on orbit results; however, the environment is reduced by one large object and its cross-sectional area, only to be replaced by a large number of smaller objects and their net target area. Although not planned, the rate of fragmentation is a direct result of human activity and is included here with the xe2x80x9cintelligentxe2x80x9d deposition of objects in the LEO regime.
Finally, the capability to retrieve objects has been demonstrated (e.g., LDEF) and has also been discussed as a possible mode of debris reduction. This component of the deposition term is negative.
In general, in the baseline model being described here, the base of LEO is taken to be that altitude at which an average member of the population has only one year left on orbit. Furthermore, only objects deposited on orbit at an altitude greater than this base and remaining there for at least one year are counted as members of the environmentxe2x80x94hereafter this requirement will simply be referred to as the xe2x80x9cmembership condition.xe2x80x9d With these provisions in mind, the expression for A is as follows:
A =L[(P1)(D1)+(FE)(DE)(PE)]xe2x88x92REMxe2x80x83xe2x80x83(2)
Where: L=launches per year, worldwide
P1=average number of pieces per launch
D1=fraction of P1 meeting membership conditions
FE=fraction of launches resulting in an on-orbit fragmentation
DE=fraction of FE meeting membership conditions
PE=average number of fragments produced per explosion
REM=number of objects retrieved per year from LEO
In the absence of a retarding medium, all objects in LEO would remain on orbit for an indefinite period of time. However, the residual atmosphere is sufficient to cause the eventual decay and reentry of some objects in this region. The efficiency of this mechanism to remove objects from orbit is dependent on the object""s altitude, orbital and physical characteristics, the phase of the solar cycle, and so on. Other factors being equal, small objects tend to be more susceptible to the action of drag forces by virtue of their (typically) larger area-to-mass ratios. In addition, the possibility of using orbital debris xe2x80x9csweepersxe2x80x9d or some equivalent process for cleaning up the orbital debris environment has been discussed.
It is assumed that some device or system is possible that may be employed to remove debris objects of all sizes, with the same efficiency, and regardless of their inherent drag characteristics. For example, such a system, when deployed, might sweep up 1% of all orbital debris objects per year. Taken together with natural decay, the B term is written as follow:
B=[Batm+S]xe2x80x83xe2x80x83(3)
Where: Batm=reduction fraction per year due to natural drag
S=reduction fraction per year due to use of debris sweeper system of some type
To determine the number of objects created per unit time due to collision, the C term is expressed as the product of two quantities shown here as follows:
C=(xcex4)H11xe2x80x83xe2x80x83(4)
Where: xcex4=number of pieces produced as a result of a collision less the two destroyed
H11=collision frequency (yrxe2x88x921) between members of a population of similar objects
The collision product factor xcex4 is obtainable from a sufficient base of experimental data or from theory. We will take xcex4 to be a constant in this simplest possible model of the environment where the population, at all times, is assumed to be made up of equivalent particles whose characteristics, overall, may change as a function of time. (A more sophisticated treatment of collision products could be allowed for if the population were partitioned into a number of different particle size regimes as will be discussed below.)
The H11 term is developed along a line of reasoning similar to that of the kinetic theory of gasses as is expressed for members of a population of similar objects as follows:                               H          11                =                                            (                              F                v                            )                        ⁡                          [                                                                    (                                                                  2                                            ⁢                                              V                        c                                                              )                                    ⁢                                      D                    1                    2                                                                                        (                                          4                      /                      3                                        )                                    ⁢                                      (                                                                  R                        T                        3                                            -                                              R                        B                        3                                                              )                                                              ]                                ⁢                      (                                          1                -                                  1                  /                                      N                    1                                                              2                        )                                              (        5        )            
Where: H11=collision frequency (yrxe2x88x921) between members of a population of similar objects
Fv=incomplete mixing factor
Vc=orbital speed at average population altitude
Dl=average population object diameter
RT=radius of the top of LEO shell from the Earth""s center
RB=radius of the base of LEO shell from the Earth""s center
Strictly speaking, the expression in Equation (5) is valid only for objects free to move at random in the specified volume like a gas. Also implicit in this formulation is the assumption that the orientation of the velocity vector of one particle with respect to all others is completely at random. It is clear that neither of these conditions is completely realized for orbital debris pieces in LEO.
The       2    ⁢      V    c  
term, for a typical LEO orbit speed, yields about 10 km/sec. However, the assumption implicit in (5) that every particle has access to all parts of the LEO volume cannot possibly be correct. The Fv term, however, is a tool whereby a phenomenological correction may be applied. An examination of orbital eccentricities is required to calculate Fv. Alternatively, one could determine Fv empirically by comparing the predicted collisions arising out of an assumption of Fv=1.0 with the actual number of collisions (if any) to date which would imply a value for Fv less than 1.0.
In a fashion similar to fragmentations, a collision between two objects results in the reduction of the total number of objects in LEO by two, along with their contribution to the total cross-sectional area for collisions. This reduction is more than compensated by the addition of the combined cross-sectional area of all of the fragments and a net increase in the total number of (smaller) objects.
The implementation of the expansion of the model to more particle sizes is straightforward, and Equation (1) extended to the case of a single stratum containing 3 species of particles (for example) is as follows:                               N          k                =                              A            k                    +                                    B              k                        ⁢                          N              k                                +                                    ∑                              i                =                1                            3                        ⁢                          xe2x80x83                        ⁢                                          ∑                                  j                  =                  1                                3                            ⁢                              xe2x80x83                            ⁢                                                δ                                                            (                      ij                      )                                        ⁢                    k                                                  ⁢                                  H                  ij                                ⁢                                  N                  i                                ⁢                                  N                  i                                                                                        (        6        )            
where the index k may take on values from 1 to N (3 for this case). Regarding the Hij factor, Equation (5) is appropriate if i=j; for dissimilar objects the appropriate form is as follows:                               H          ij                =                              (                          F                              v                ij                                      )                    ⁢                      {                                                                                                                              (                                                                              2                                                    ⁢                                                      V                            c                                                                          )                                            [                                                                        D                          i                                                +                                                  D                          1                                                                    )                                        /                    2                                    ]                                2                                                              (                                      4                    /                    3                                    )                                ⁢                                  (                                                            R                      T                      3                                        -                                          R                      B                      3                                                        )                                                      }                                              (        7        )            
All variables in Equations (6) and (7) are defined in fashions analogous to Equations (1) and (5). Collisions between two objects, in general, result in a decrease in the number of large objects while increasing the number of small objects such that mass conservation is maintained.
The model is implemented by the numerical solution of the set of coupled differential equations described by equations (6) and (7), for the most general case of multiple debris sizes and multiple strata. The solution of the equation set may be accomplished using standard differential equation solvers such as Euler methods or Runge-Kutta procedures. The following includes, but is not limited to, the capabilities of the model:
1. Environmental evolutionxe2x80x94near term and long term
2. Modeling of solar effects on the environment
3. Environment response to policy implementation and mitigation
Launch rate variation
Operational debris limitation
Procedures to limit explosions and breakups
Efficacy of collision-avoidance procedures
4. Mitigation
Large debris removal (rocket bodies)
Small debris removal (sweepers, laser removal, etc.)
5. Environment response to catastrophe
Explosion and Breakups
Sabotage and Warfare
6. High-value-asset concerns
International Space Station risk assessment against nominal and special environments
Constellation risk assessment against nominal and special environments
Cumulative exposure risk to astronauts on EVA
Other individual or group assets (satellite, space telescope, space laboratory, etc.)
Asset risk assessment based on collision rate
Collision ratexe2x80x94nominal
Collision ratexe2x80x94temporary elevation due to nearby explosions and breakups
Use of risk assessment to calculate insurance for high-value assets
The technology can be used as a stand-alone software package or via the Internet as an application service provider (ASP) service. For sophisticated applications, the program can be implemented by an expert group performing as consultants.